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MMLU-Pro / 8851 / Use divergence therem to evaluate iint_S vec{F} cdot d vec{S} where…

Problem

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question

Use divergence therem to evaluate $\iint_S \vec{F} \cdot d \vec{S}$ where F→=yx2i→+(xy2−3z4)j→+(x3+y3)k→\vec{F} = yx^2 \vec{i} + (xy^2 - 3z^4)\vec{j} + (x^3+y^3)\vec{k} and the surface SS consists of the sphere of radius 4 with z≤0z \le 0 and y≤0y \le 0. Note all three surfaces of this solid are included in SS.
Plain-text mathematical notation (without MathML)
Use divergence therem to evaluate $\iint_S \vec{F} \cdot d \vec{S}$ where (F)→=yx²(i)→+(xy²−3z⁴)(j)→+(x³+y³)(k)→ and the surface S consists of the sphere of radius 4 with z≤0 and y≤0. Note all three surfaces of this solid are included in S.
Original LaTeX notation
Use divergence therem to evaluate $\iint_S \vec{F} \cdot d \vec{S}$ where $\vec{F} = yx^2 \vec{i} + (xy^2 - 3z^4)\vec{j} + (x^3+y^3)\vec{k}$ and the surface $S$ consists of the sphere of radius 4 with $z \le 0$ and $y \le 0$. Note all three surfaces of this solid are included in $S$.

category

math

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src

theoremQA-Math

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Source and history

upstream_id / Original problem ID
8851
task_id
41a3ce43-86cb-59d9-a190-ee7bffd4ed53
task_key
default--test--8851
task_revision_id
1
config / split
default / test

Official source

initial import